IB Math Studies – Functions

IB Math Studies Functions

Questions and Answers

1. IB Mathematical Studies SL – Functions, simplest form

How can I find in simplest form $f(-x), f(x-3), f(2x+1)$ ,where $f(x)=2x^2-3x$ .

Solution

IB Mathematical Studies SL – Functions, simplest form

The functions in simplest form are:

$f(-x)= 2(-x)^2-3(-x)=2x^2+3x$

$f(x-3)= 2(x-3)^2-3(x-3)=2(x^2-6x+9)-3x+9=2x^2-12x+18-3x+9=2x^2-15x+27$

$f(2x+1)= 2(2x+1)^2-3(2x+1)=2((2x)^2+4x+1)-6x-3=$

$= 8x^2+8x+2-6x-3=8x^2+2x-1$

2. IB Mathematical Studies SL – Linear Functions

How can we find a formula that describes the cost of running a car if there is a standard cost of $100 plus $0.1 per one kilometer thereafter?

Solution

IB Mathematical Studies SL – Linear Functions

Let x be the kilometers then the linear function that describes the above word problem is:

$f(x)= 100+0.1x$

3. IB Mathematical Studies SL – Quadratic Functions

The profit of a PC manufacturer company of making x personal computers (PC) per day is given by the following formula:

$f(x)=-0.2x^2+12x+220$

How can we find the profit in dollars in the following cases:

for 10 PCs, 40PCs and 100PCs

Solution

IB Mathematical Studies SL – Quadratic Functions

The profit of the company when being produced 10 PCs is given by

$f(10)=-0.1(10^2)+12(10)+220=-10+120+220=$330$

The profit of the company when being produced 40 PCs is given

$f(40)=-0.1(40^2)+12(40)+220=-160+480+220=$540$

The profit of the company when it produces 100 PCs is given

$f(40)=-0.1(100^2)+12(100)+220=-1000+1200+220=$420$

4. IB Mathematical Studies SL – Complete the square

How can we complete the square and find the vertex of the following quadratic function:

$f(x)=3x^2-9x+15$

Solution

IB Mathematical Studies SL – Complete the square

Completing the square is a procedure for reformatting quadratic functions.

Quadratic Functions of the form $f(x)=ax^2+bx+c$ can be rewritten as

$f(x)=a(x-h)^2+k$

The major reason for doing this is that it allows us to easily see the vertex of the curve $f(x)=ax^2+bx+c$ which is located at the point $(h,k)$ .

The general method of completing the square can be shown like this:

$f(x)=ax^2+bx+c=a(x^2+\frac{b}{a}x)+c=$

$=a((x+\frac{b}{2a})^2-(\frac{b}{2a})^2)+c=$

$=a(x-(-\frac{b}{2a}))^2-\frac{b^2}{4a}+c$

Therefore the x-coordinate of the vertex is $h=-\frac{b}{2a}$

Concerning this question if we follow the preceding procedure we have:

$f(x)=3x^2-9x+15=3(x^2+\frac{-9}{3}x)+15=$

$=3((x+\frac{-9}{2(3)})^2-(\frac{-9}{2(3)})^2)+15=$

$=3(x-(-\frac{-9}{2(3)}))^2-\frac{(-9)^2}{12}+15=$

$=3(x-(\frac{3}{2}))^2-\frac{81}{12}+15=$

$=3(x-(\frac{3}{2}))^2-\frac{81}{12}+\frac{15(12)}{12}=$

$=3(x-(\frac{3}{2}))^2-\frac{81}{12}+\frac{180}{12}=$

$=3(x-(\frac{3}{2}))^2+\frac{99}{12}=$

Therefore the coordinates of the vertex are $( \frac{3}{2}, \frac{99}{12})$

5. IB Mathematical Studies SL – Factorisation

How can we factorize this expression $4x^2-16$ ?

Solution

IB Mathematical Studies SL – Factorization, difference of squares

We‘ll use the difference of squares identity

$a^2-b^2=(a-b)(a+b)$

Concerning this question the factorization is

$4x^2-16=(2x)^2-4^2=(2x-4)( 2x+4)$

6. IB Mathematical Studies SL – Factorisation, Common Factor

How can we fully factorize the following expression:

$(3x-4)^2-9(3x-4)$ ?

Solution

IB Mathematical Studies SL – Factorisation, common factor

The expression can be factorized as follows

$(3x-4)^2-9(3x-4) =(3x-4)((3x-4)-9)= (3x-4)(3x-13)$ ?